Hermite–Hadamard–Fejér type inequalities involving generalized fractional integral operators

Abstract

Since the so-called Hermite–Hadamard type inequalities for convex functions were presented, their generalizations, refinements, and variants involving various integral operators have been extensively investigated. Here we aim to establish several Hermite–Hadamard–Fejér type inequalities for symmetrized convex functions and Wright-quasi-convex functions with a weighted function symmetric with respect to the midpoint axis on the interval involving the known generalized fractional integral operators. We also point out that certain known inequalities are particular cases of the results presented here.

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References

  1. 1.

    Agarwal, R.P., M.-J. Luo, and R.K. Raina. 2016. On Ostrowski type inequalities. Fasciculi Mathematici 204: 5–27.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bombardelli, M., and S. Varošanec. 2009. Properties of \(h\)-convex functions related to the Hermite-Hadamard-Fejér inequalities. Computers and Mathematics with Applications 58: 1869–1877.

    Google Scholar 

  3. 3.

    Dahmani, Z. 2010. New inequalities in fractional integrals. International Journal of Applied Nonlinear Science 9 (4): 493–497.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Dragomir, S.S. 2016. Symmetrized convexity and Hermite-Hadamard type inequalities. Journal of Mathematical Inequalities 10 (4): 901–918.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dragomir, S.S. 2017. Some inequalities of Hermite-Hadamard type for symmetrized convex functions and Riemann-Liouville fractional integrals. RGMIA Res. Rep. Coll 20: 15. (Art. 46).

    Google Scholar 

  6. 6.

    Dragomir, S.S., and C.E.M. Pearce. 2000. Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs. Footscray: Victoria University.

    Google Scholar 

  7. 7.

    Dragomir, S.S., and C.E.M. Pearce. 1998. Quasi-convex functions and Hadamard’s inequality. Bulletin of the Australian Mathematical Society 57 (3): 377–385.

    MathSciNet  Article  Google Scholar 

  8. 8.

    El Farissi, A., M. Benbachir, and M. Dahmane. 2012. An extension of the Hermite-Hadamard inequality for convex symmetrized functions. Real Analysis Exchange 38 (2): 467–474.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fejér, L. 1906. Uberdie Fourierreihen \(\prod\). Math. Naturwise. Anz Ungar. Akad., Wiss 24: 369–390. (in Hungarian).

    Google Scholar 

  10. 10.

    Gorenflo, R., and F. Mainardi. 1997. Fractional Calculus: Integral and Differential Equations of Fractional Order, 223–276. Wien: Springer.

    Google Scholar 

  11. 11.

    İşcan, İ. 2014 Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals, arXiv preprint arXiv:1404.7722.

  12. 12.

    Mitrinović, D.S., and I.B. Lacković. 1985. Hermite and convexity. Aequationes Mathematicae 28: 229–232.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Raina, R.K. 2005. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Mathematical Journal 21 (2): 191–203.

    MATH  Google Scholar 

  14. 14.

    Sarıkaya, M.Z., E. Set, H. Yaldız, and N. Başak. 2013. Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling 57 (9): 2403–2407.

    Article  Google Scholar 

  15. 15.

    Sarıkaya, M.Z., and H. Yıldırım. 2016. On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals. Miskolc Mathematical Notes 17 (2): 1049–1059.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Set, E., A.O. Akdemir, and B. Çelik. On Generalization of Fejér type inequalities via fractional integral operator, Filomat (accepted).

  17. 17.

    Set, E., B. Çelik, and A.O. Akdemir. 2017. Some new Hermite-Hadamard type inequalities for Quasi-convex functions via fractional integral operator. AIP Conference Proceedings 1833 (1): 020021.

    Article  Google Scholar 

  18. 18.

    Set, E., and B. Çelik. 2018. Generalized fractional Hermite-Hadamard type inequalities for \(m\)-convex and (\(\alpha\), \(m\))-convex functions. Communications Faculty of Sciences University of Ankara Series 67 (1): 351–362. Series A1.

    Google Scholar 

  19. 19.

    Set, E., and B. Çelik. On generalizations related to the left side of Fejér’s inequality via fractional integral operator, Miskolc Mathematical Notes, (accepted).

  20. 20.

    Set, E., J. Choi, and B. Çelik. 2017. Certain Hermite-Hadamard type inequalities involving generalized fractional integral operators. RACSAM. https://doi.org/10.1007/s13398-017-0444-1.

  21. 21.

    Set, E., J. Choi, and A. Gözpınar. 2017. Hermite-Hadamard type inequalities for the generalized \(k\)-fractional integral operators. Journal of Inequalities and Application. https://doi.org/10.1186/s13660-017-1476-y (Article ID 206).

  22. 22.

    Set, E., and A. Gözpınar. 2017. Some new inequalities involving generalized fractional integral operators for several class of functions. AIP Conference Proceedings 1833 (1): 020038.

    Article  Google Scholar 

  23. 23.

    Set, E., and A. Gözpınar. 2017. Hermite-Hadamard Type Inequalities for convex functions via generalized fractional integral operators. Journal of Inequalities and Applications. https://doi.org/10.1186/s13660-017-1476-y.

    Article  MATH  Google Scholar 

  24. 24.

    Set, E., Gözpınar, A., and E.A. Alan. 2018. Generalized fractional integral inequalities for some classes of symmetrized convex functions. AIP Conference Proceedings 1991: 020010. https://doi.org/10.1063/1.5047883.

    Article  Google Scholar 

  25. 25.

    Set, E., İ. İşcan, M.Z. Sarıkaya, and M.E. Özdemir. 2015. On new inequalities of Hermite-Hadamard-Fejér type for convex functions via fractional integrals. Applied Mathematics and Computation 259: 875–881.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Set, E., M.A. Noor, M.U. Awan, and A. Gözpınar. 2017. Generalized Hermite-Hadamard type inequalities involving fractional integral operators. Journal of Inequalities and Applications. https://doi.org/10.1186/s13660-017-1444-6 (Article ID 169).

  27. 27.

    Set, E., M.Z. Sarıkaya, M.E. Özdemir, and H. Yıldırım. 2014. The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results. Journal of Applied Mathematics, Statistics and Informatics 10 (2): 69–83.

    MathSciNet  Article  Google Scholar 

  28. 28.

    Srivastava, H.M., and J. Choi. 2012. Zeta and q-Zeta Functions and Associated Series and Integrals. Amsterdam: Elsevier Science Publishers.

    Google Scholar 

  29. 29.

    Varošanec, S. 2007. On \(h\)-convexiety. Journal of Mathematical Analysis and Applications 326 (1): 301–311.

    Google Scholar 

  30. 30.

    Yaldız, H., and M.Z. Sarıkaya. On the Hermite-Hadamard type inequalities for fractional integral operator, Preprint.

  31. 31.

    Usta, F., H. Budak, M.Z. Sarıkaya, and E. Set. On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat (accepted).

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Acknowledgements

This research is supported by Ordu University Scientific Research Projects Coordination Unit (BAP). Project Number: B-1809.

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Correspondence to Junesang Choi.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Erhan Set declares that he has no conflict of interest. Junesang Choi declares that he has no conflict of interest. E. Aykan Alan declares that he has no conflict of interest.

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Set, E., Choi, J. & Alan, E.A. Hermite–Hadamard–Fejér type inequalities involving generalized fractional integral operators. J Anal 27, 1007–1027 (2019). https://doi.org/10.1007/s41478-018-0159-5

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Keywords

  • Convex function
  • Quasi-convex function
  • Symmetrized convex function
  • Wright-quasi-convex functions
  • Hermite–Hadamard type inequalities
  • Generalized fractional integral operators
  • Hermite–Hadamard–Fejér type inequalities

Mathematics Subject Classification

  • 26A33
  • 26D10
  • 26D15
  • 33B20